Vol. 7, No. 2, November 2010, 253-268
Robust State Feedback Controller Design of
STATCOM Using Chaotic Optimization Algorithm
Amin Safari
1, Hossein Shayeghi
2, Heidar Ali Shayanfar
3Abstract: In this paper, a new design technique for the design of robust state feedback controller for static synchronous compensator (STATCOM) using Chaotic Optimization Algorithm (COA) is presented. The design is formulated as an optimization problem which is solved by the COA. Since chaotic planning enjoys reliability, ergodicity and stochastic feature, the proposed technique presents chaos mapping using Lozi map chaotic sequences which increases its convergence rate. To ensure the robustness of the proposed damping controller, the design process takes into account a wide range of operating conditions and system configurations. The simulation results reveal that the proposed controller has an excellent capability in damping power system low frequency oscillations and enhances greatly the dynamic stability of the power systems. Moreover, the system performance analysis under different operating conditions shows that the phase based controller is superior compare to themagnitude based controller.
Keywords: STATCOM, State feedback damping controller, Chaotic optimiza-tion algorithm, Low frequency oscillaoptimiza-tions.
Nomenclature
COA Chaotic Optimization Algorithm STATCOM Static Synchronous Compensator
DC Direct Current
SVC Static Var Compensator
q
E′ Internal voltage behind transient reactance
A
T Regulator time constant
fd
E Equivalent excitation voltage
0
d
T′ Time constant of excitation circuit
FACTS Flexible Alternating Current Transmission Systems
1Young Researchers Club, Islamic Azad University, Ahar branch, Iran; E-mail: asafari1650@ yahoo.com 2Department of Technical Engineering, University of Mohaghegh Ardabili, Ardabil, Iran.
3Center of Excellence for Power System Automation and Operation, Electrical Engineering Department, Iran
University of Science and Technology, Tehran, Iran
e
T Electric torque GTO Gate Turn Off Thyristor
dc
V DC capacitor voltage
ITAE Integral of Time multiplied Absolute value of the Error
t
V Terminal voltage
A
K Regulator gain
ref
v Reference voltage
M Machine inertia coefficient VSC Voltage Source Converter
e
P Electrical output power
ω
Rotor
speed
PI Proportional integral
δ
Rotor
angle
m
P Mechanical input power
ϕ Excitation phase angle
SDT Step down transformer
e
P
Δ Electrical power deviation SMIB Single Machine Infinite Bus
dc
V
Δ DC voltage deviation
1 Introduction
Intensive progress in power electronics has enabled the application of Flexible AC Transmission System (FACTS) devices in high voltage transmission networks. The main objective of using FACTS devices are normally the steady-state control of a power system, but due to their fast response, FACTS can also be used for power system stability enhancement by improving damping of the power swings [1]. Through the modulation of bus voltage, phase shift between buses, and transmission line reactance, FACTS devices can cause a substantial increase in power transfer limits during the steady-state conditions.
control stability, lower harmonics and smaller size, etc. [2]. Several trials have been reported in the literature to dynamic models of the STATCOM in order to design suitable controllers for AC, DC voltage and damping controls. Wang [3] presents the establishment of the linearized Phillips-Heffron model of a power system installed with a STATCOM. Further, no effort seems to have been made to identify the most suitable STATCOM control parameter, in order to arrive at a robust damping controller. Fuzzy-logic-based controllers have, for example, been used for controlling a STATCOM [4]. The performance of such controllers can further be improved by adaptively updating their parameters. Also, although by using the robust control methods [5, 6], the uncertainties are directly introduced to the synthesis, but due to the large model order of the power systems the resulting controller order will be very large in general, which is not feasible because of the computational difficulties and economical expenses in implementation.
In this paper, COA is used for robust tuning of the STATCOM based state feedback damping controller in order to enhance the damping of the power systems low frequency oscillations. Chaos is a kind of characteristic of nonlinear systems which is a bounded unstable dynamic behavior, which exhibits sensitive dependence on the initial conditions and includes infinite unstable periodic motions. The COA is based on the ergodicity, stochastic properties and regularity of the chaos. It is not like some stochastic optimization algorithms that escape from the local minima by accepting some bad solutions according to a certain probability, but COA searches on the regularity of the chaotic motion to escape from the local minima [7-13].
A new approach for the optimal design of state feedback gains for the STATCOM damping controller is investigated in this paper. The problem of robust state feedback damping controller design for STATCOM is formulated as an optimization problem and COA is used to solve it. Since only local states are used as the inputs of each controller, the optimal design of the controller can be accomplished. The effectiveness of the proposed controller is demonstrated through nonlinear time-domain simulation studies to damp low frequency oscillations under different operating conditions. Results evaluation show that the proposed damping controller achieves good robust performance for a wide range of operating conditions and disturbances.
2
System Model with STATCOM
The synchronous generator is delivering power to the infinite-bus through a double circuit transmission line and a STATCOM. The system data is given in the Appendix. The system consists of a step down transformer with a leakage reactance XSDT, a three phase GTO-based voltage source converter, and a dc capacitor [3].
Vt
Vd c Vb
c f V S C V
E t L 1 V m
L 2
L 3
L 4
Fig. 1 – SMIB power system equipped with STATCOM.
2.1 Power system nonlinear model with STATCOM
The VSC generates a controllable AC voltage source v to( )=V0sin(ω − ϕt ) behind the leakage reactance. The voltage difference between the STATCOM bus AC voltage, v tL( ) and v t0( ) produces active and reactive power exchange between the STATCOM and the power system, which can be controlled by adjusting the magnitude V0 and the phase angle ϕ. The dynamic relation between the capacitor voltage and current in the STATCOM circuit are asserted as [3]:
j
Lo Lod Loq
I =I + I , (1)
(cos jsin )
o dc dc
V =cV ϕ + ϕ =cV ∠ϕ, (2)
( cos sin )
dc
dc Lod Loq
dc dc
I c
V I I
C C
= = ϕ + ϕ , (3)
where, for the PWM inverter c=mk and k is the ratio between AC and DC voltage depending on the inverter structure, m and c are the modulation ratio and magnitude defined by the PWM. The Cdc is the dc capacitor value and Id0
is the capacitor current while iL d0 and iL q0 are the d and q components of the STATCOM current, respectively.
The dynamics of the generator and the excitation system are expressed through a fourth order model given as [5, 8, 16]:
0( 1)
δ = ω ω − (4)
(Pm Pe D ) /M
.
0
( ) /
q q fd d
E′= −E +E T′ (6)
.
( ( )) /
fd fd a ref t a
E = −E +K V −V T (7)
The expressions for the d-q axes currents in the transmission line and STATCOM are given as follows [5]:
1 sin cos
1
LB LB
q dc b
SDT SDT
tld
tL LB
tL LB d
LB SDT
X X
e mV V
X X
I
X X
X X x
X X
⎛ ⎞ ′
+ − ϕ − ϕ
⎜ ⎟ ⎝ ⎠ = ⎛ ⎞ ′ + + + +⎜ ⎟ ⎝ ⎠ , (8) cos sin 1 LB dc b SDT tlq tL LB
tL LB q
LB SDT X mV V X I X X
X X x
X X
ϕ + ϕ
=
⎛ ⎞
+ + + +⎜ ⎟
⎝ ⎠
, (9)
( ) sin
q d tL tLq dc
Lod
SDT
e x X I mV
I
X
′ − ′ + − ϕ
= , (10)
cos ( )
dc d tL tLq
Loq
SDT
mV x X I
I
X ′ ϕ − +
= , (11)
where, XtL =XT +XL/ 2, / 2XLB= XL .
The XT, xd′ and xq are the transmission line reactance, d-axis transient reactance, and q-axis reactance, respectively.
2.2 Power system linearized model
A linear dynamic model is obtained by linearizing the nonlinear model around an operating point. The linearized model of the power system as shown in Fig. 1 is given as:
0
Δδ = ω Δω, (12)
( Pe D ) /M
Δω = −Δ − Δω , (13)
0
( ) /
q q fd d
E′ E E T′
Δ = −Δ + Δ , (14)
( ( ) ) /
fd A ref fd A
E K v v E T
Δ = Δ − Δ − Δ , (15)
7 8 9
dc q dc dc d
v K K E′ K v K c K ϕ
Δ = Δδ + Δ − Δ + Δ + Δϕ, (16)
1 2
e q pdc dc pc p
P K K E′ K v K c K ϕ
4 3
q q qdc dc qc q
E′ K K E′ K v K c K ϕ
Δ = Δδ + Δ + Δ + Δ + Δϕ, (18)
5 6
t q vdc dc vc v
V K K E′ K v K c Kϕ
Δ = Δδ + Δ + Δ + Δ + Δϕ, (19)
where, K1, K2…,K9, Kpu, Kqu and Kvu are linearization constants. The state-space model of the power system is given by:
x=Ax+Bu, (20)
where, the state vector x, control vector u, A and B are:
[ ]T
q fd dc
E′ E v
= Δδ Δω Δ Δ Δ
x ; u= Δ
[
c Δϕ]
T,0
1 2
3 4
0 0 0 0
5 6
7 8 9
0 0
0 0 0 0
0 0 1 0 ; 1 0 0 0 pc p pdc qc q qdc do do
d d d d
A v
A A A vdc A vc
A A A A A A
dc d w K K K K K M M
M M M
K K
K K
K
T T
T T T T
K K
K K K K K K K K
T T T T T T
K K K K K
ϕ ϕ ϕ ϕ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ − − − ⎥ − − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ − − − − ⎥ − − =⎢ ′ ′ ′ ′ ⎥ = ⎢ ′ ′ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢− − − − ⎥ − − ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦
A B .
The block diagram of the linearized model of the SMIB power system with the STATCOM is shown in Fig. 2.
2.3 State feedback damping controller for the STATCOM
The state feedback based damping controller is designed to produce an electrical torque in phase with the speed deviation according to the phase compensation method. A power system can be modeled by a set of nonlinear differential equations as:
( , )
f =
X X U , (21)
When a power system operates at one operating point, the system is linearized. The state-space model of power system is given by:
= +
x Ax Bu, (22)
where the input vector u is composed of the control vector
S
U and the
disturbance vector Ud, both m×1 vectors; x is an n×1 state vector; A is an
n n× plant matrix of the open-loop system and B is the n m× input matrix; n
and m are the number of state variables and control signals, respectively. In this paper C and ϕ are modulated in order to use it for the state feedback controller design. The following control input vector is defined:
s= x
U K , (23)
where, K is the feedback gain matrix with appropriate dimensions. Applying
(23) to (22):
( ) d
= + +
x A BK x BU , (24)
c = +
A A BK, (25)
where, AC is a the closed-loop system matrix. By properly choosing the
feedback gain K, the eigenvalues of the closed-loop matrix
C
A are moved to
the left-hand side of the complex plane and the desired performance of the controller can be achieved. Thus, the remaining problem in the design of the output feedback controller is the selection of K to achieve the required
objectives. The control objective is to increase the damping of the critical modes to the desired level.
3
Chaotic Optimization Algorithm
systems containing multiple elements with nonlinear interactions, particularly when the system is forced and dissipative. Sensitive dependence on the initial conditions is not only observed in the complex systems, but even in the simplest logistic equation [10]. The application of the chaotic sequences can be an interesting alternative to provide the search diversity in an optimization procedure. Due to the non-repetition of the chaos, it can carry out the overall searches at higher speeds than the stochastic ergodic searches which depend on the probabilities. The basic process of the chaos optimization algorithm generally includes two major steps. Firstly, defines a chaotic sequences generator based on the Lozi map. Generates a sequence of chaotic points and map it to a sequence of the design points in the original design space. Then, calculate the objective function with respect to the generated design points and chooses the point with the minimum objective function as the current optimum. Secondly, the current optimum is assumed to be close to the global optimum after certain iterations and it is viewed as the center with a little chaotic perturbation and the global optimum is obtained through the fine search. Repeat the above two steps until some specified convergence criterion is satisfied and then the global optimum is obtained [12]. This chaotic map involves also non-differentiable functions which make the modeling of the associate time series difficult. The Lozi map is given by [9, 14]:
1( ) 1 1( 1) ( 1)
y k = − ×a y k− +y k− , (26)
1
( ) ( 1)
y k = ×b y k− , (27)
( ) ( ) y k
z k = − α
β − α , (28)
( ) ( ) , 0.5,
( ) ( ) , 0.5,
i i i i i
i i i i i
x k x z k U X r
x k x z k X L r
= + λ × × − ≤
= − λ × × − > (29)
where, k is the iteration number. The values of y are normalized in the range [0, 1] to each decision variable in the n-dimensional space of optimization problem. Therefore, y1∈ −[ 0.6418, 0.6716] and [ , ]α β = −( 0.6418,0.6716). The parame-ters used in this work are a=1.7 and b=0.5, as proposed by [11]. Many, unconstrained optimization problems with continuous variables can be formulated as the following functional optimization problem:
Find X to minimize (f X), X =[ ,x x1 2,…,xn],
where, f is the objective function, and X is the decision solution vector
[14]. Fig. 3 shows the flowchart of the proposed chaotic search procedure based on the Lozi map [9].
g
M and ML are maximum number of iterations of the chaotic global
search and maximum number of iterations of the chaotic local search, respectively. In this paper λ is the step size in chaotic local search and linearly decreases from 0.1 to 0.01. Also, f and X are the best objective function and
the best solution for the state feedback controller gains from the current run of the chaotic search, respectively.
Fig. 3 – Flowchart of the proposed COA.
4
Design of the State Feedback Controllers Using COA
problem. In this study, an Integral of Time multiplied Absolute value of the Error (ITAE) is taken as the objective function. Since the operating conditions in power systems are often varied, a performance index for a wide range of operating points is defined as [8]:
1 0
d
sim
t Np
i i
f t t
=
=
∑ ∫
Δω , (30)where, tsim is the time range of simulation and Np is the total number of
operating points for which the optimization is carried out. For the objective function calculation, the time-domain simulation of the power system model is carried out for the simulation period. The design problem can be formulated as the following constrained optimization problem, where the constraints are the controller parameters bounds:
Minimize f Subject to: min max, 1,...,5
x x x
G ≤G ≤G x= . (31)
The proposed approach employs COA to solve this optimization problem and search for an optimal set of output feedback controller gains. In any power system, the loading condition varies over a wide range. It is extremely essential to investigate the effect of variation of the loading condition on the dynamic performance of the system. In order to test the robustness of the state feedback controllers to wide variation in the loading condition, the uncertainty area for active and reactive power are selected as: 0.2≤ ≤P 1.2 and 0.01≤ ≤Q 0.4 pu [15]. The optimization of controller gains is carried out by evaluating the objective function as given in equation (30), which considers multiple of operating conditions. The operating conditions are considered as:
•Base case: P=0.80pu, 0.2Q= pu and XL =0.4pu (Nominal loading).
•Case 1: P=0.2pu, 0.01Q= pu and XL =0.4 pu (Light loading).
•Case 2: P=1.2pu, 0.4Q= and XL =0.4pu (Heavy loading).
•Case 3: The 20% increase of line reactance XL at nominal loading condition.
•Case 4: The 20% increase of line reactance XL at heavy loading condition.
search and linearly decreases from 0.1 to 0.01. It should be noted that the COA is run several times and then the optimal set of the controller parameters is selected. The final values of the optimized parameters are given in Table 1.
Table 1
Optimal parameters of the controller.
Controller Parameter
C Based Controller
φ Based Controller
G1
245.71184.56
G2 1.08767.3446
G3 0.05410.3445
G4 0.87530.3876
G5
0.9588-1.565
5
Nonlinear Time Domain Simulation
To investigate the power system performance, two classes of disturbances are studied. These classes are chosen to represent large, as well as small power system disturbances.
5.1 Response to small disturbance
To assess the performance of the proposed controllers, a disturbance of 0.3pu input torque is applied to the machine at t=1s. The response of the system without controller is shown in Fig. 4.
The study is performed at three different operating conditions. The results are shown in Fig. 5. It can be seen that the COA based designed state feedback controller achieves good robust performance and enhance greatly the dynamic stability of the power systems.
5.2 Response to large disturbance
of the state feedback controllers are shown in Fig. 7 at nominal loading conditions. It can be concluded that the ϕ-based controller provides much less control effort compared to the C-based controller.
Fig. 4 –Response of the system without controller.
Fig. 5a – Dynamic responses for Δω at nominal loading; Solid (COA based φ controller) and Dashed (COA based C controller).
Fig. 5b – Dynamic responses for Δω at light loading;
Fig. 5c – Dynamic responses forΔωat heavy loading;
Solid (COA basedφcontroller) and Dashed (COA based C controller).
Fig. 6a – Dynamic responses forΔωatnominal loading;
Solid (COA basedφcontroller) and Dashed (COA based C controller).
Fig. 6b – Dynamic responses forΔωatlight heavy loading;
Fig. 6c – Dynamic responses forΔωatheavy loading;
Solid (COA basedφcontroller) and Dashed (COA based C controller).
Fig. 7 – Control signals of both stabilizers for a large disturbance at nominal loading;
Solid (COA basedφcontroller) and Dashed (COA based C controller).
6 Conclusion
7 Appendix
The nominal parameters of the system are listed in Table 2.
Table 2 System parameters. 8 MJ/MVA
=
M Tdo′ =5.044s Xd =1pu
Generator
0.6 pu
=
q
X Xd′ =0.3pu D=0
Excitation system Ka =25 Ta =0.05s
Transformers XT =0.1pu XSDT =0.1pu
DC link parameter VDC =1pu CDC =1pu
0.25
=
C ϕ =52°
STATCOM parameter
1
= s
K Ts =0.05
8 References
[1] J. Machowski, J.W. Bialek: State Variable Control of Shunt FACTS Devices using Phasor Measurements, Electrical Power System Research. Vol. 78, No. 1, Jan. 2008, pp. 39 – 48. [2] N.G. Hingorani, L. Gyugyi: Understanding FACTS: Concepts and Technology of Flexible
AC Transmission Systems, Wiley-IEEE Press, 1999.
[3] H.F Wang: Phillips-Heffron Model of Power Systems Installed with STATCOM and Applications, IEE Proceedings Generation, Transmission and Distribution, Vol. 146, No. 5, Sept. 1999, pp. 521 – 527.
[4] S. Moris, P.K. Dash, K.P. Basu: A Fuzzy Variable Structure Controller for STATCOM, Electrical Power System Research, Vol. 65, No. 1, Apr. 2003, pp. 23 – 34.
[5] A.H.M.A. Rahim, M.F Kandlawala: Robust STATCOM Voltage Controller Design using Loop-shaping Technique, Electrical Power System Research, Vol. 68, No.1, Jan. 2004, pp. 61 – 74.
[6] F. Armansyah, N. Yorino, H. Sasaki: Robust Synchronous Voltage Sources Designed Controller for Power System Oscillation Damping, Electrical Power and Energy Systems, Vol. 24, No.1, Jan. 2002, pp. 41 – 49.
[7] H. Shayeghi, S. Jalilzadeh, H.A. Shayanfar, A. Safari: Robust PSS Design using Chaotic Optimization Algorithm for a Multimachine Power System, ECTI-CON 2009, Pattaya, Thailand, May 2009, pp. 40 – 43.
[8] H. Shayeghi , H.A. Shayanfar, S. Jalilzade, A. Safari: A PSO based Unified Power Flow Controller for Damping of Power System Oscillations, Energy Conversion and Management, Vol. 50, No. 10, Oct. 2009, pp. 2583 – 2592.
[9] L.D.S. Coelho: Tuning of PID Controller for an Automatic Regulator Voltage System using Chaotic Optimization Approach, Chaos, Solitons and Fractals, Vol. 39, No. 4, Feb. 2009, pp. 1504 – 1514.
[11] R. Caponetto, L. Fortuna, S. Fazzino, M.G Xibilia: Chaotic Sequences to Improve the Performance of Evolutionary Algorithms, IEEE Transaction on Evolutionary Computation, Vol. 7, No. 3, June 2003, pp. 289 – 304.
[12] D. Yang, G. Li, G. Cheng: On the Efficiency of Chaos Optimization Algorithms for Global Optimization, Chaos, Solitons and Fractals, Vol. 34, No. 4, Nov. 2007, pp. 1366 – 1375. [13] B. Liu, L. Wang, Y.H. Jin, F. Tang, D.X Huang: Improved Particle Swarm Optimization
Combined with Chaos, Chaos, Solitons and Fractals, Vol. 25, No. 5, Sept. 2005, pp. 1261 – 1271. [14] H. Shayeghi, S. Jalilzadeh, H. A. Shayanfar, A. Safari: Multi-machine Power System
Stabilizers Design using Chaotic Optimization Algorithm, Energy Conversion and Management, Vol. 51, No. 7, July 2010, pp. 1572 – 1580.
[15] A. Jeevanandham, K.G. Thanushkodi: Robust Design of Decentralized Power System Stabilizers using Meta-heuristic Optimization Techniques for Multimachine Systems, Serbian Journal of Electrical Engineering, Vol. 6, No.1, May 2009, pp. 89 – 103.
